linear-algebra

[tough remnant]

do i just put inverse A where A is supposed to be?

because its a different equation

or do i multiply the answer by 1/determinant

[dusky epoch]

no

from $Ax = b$, multiplying on the left by $A^{-1}$ on both sides gives you $$A^{-1}Ax = A^{-1}b$$

[stoic python]

Ann:

[dusky epoch]

which simplifies to $x = A^{-1}b$

[stoic python]

Ann:

[tough remnant]

ahh I see

thanks

but doesnt Axinverse A equal to I

[dusky epoch]

that's the point

but uh

please don't use x as multiplication

[tough remnant]

do we just ignore I in the left hand side then

[dusky epoch]

no

I is the identity matrix

in some sense it plays the role of the number 1 in the matrix world

[tough remnant]

I see

thanks

sorry for asking a lot of questions trying to solidify my understanding

[alpine echo]

https://www.reddit.com/r/3Blue1Brown/comments/elm05r/should_a100_c_d100_c1_instead_of_a100_c1_d100_c/?utm_medium=android_app&utm_source=share

Can someone please take a look at this question?

Is my arguement right ?

[dusky epoch]

can you state the timestamp at which Grant says $A^{100} = C^{-1}D^{100}C$?

[stoic python]

Ann:

[dusky epoch]

i don't want to sift through the entire video

@alpine echo

[alpine echo]

One sec

Oh wait, he doesn't actually explicitly write it out, but when he shows us a small recap(15:20) he does C^-1 A C. And then shows us that computation of 100th power is easy.

So he meant, what I think?

[dusky epoch]

okay so it's at 15:20

gimme a sec

he does $C^{-1}AC = D$

[stoic python]

Ann:

[dusky epoch]

which is equivalent to $D = CAC^{-1}$

[stoic python]

Ann:

[alpine echo]

What, I don't understand the second one

🤔

C has eigen vectors(,written in our language) as columns, which means it transforms eigen basis to our basis

Am I right ?

@dusky epoch

[dusky epoch]

the equivalence i described has a purely algebraic explanation

brb

[alpine echo]

Man, please make the explanation 3b1b linear transform logic.

C has eigen vectors(written in our language) as columns, which means it transforms eigen basis to our basis

[dusky epoch]

i mean let's put it this way

[alpine echo]

Am I right ?

[dusky epoch]

not quite.

if e_i is the i'th standard basis vector, Ce_i is the i'th vector in the eigenbasis

so if the input of C is a standard basis vector, then its output is an eigenvector

er.

wait.

i might be getting confused at this shit myself

[alpine echo]

https://www.reddit.com/r/3Blue1Brown/comments/el81cs/little_confused_by_this_eigenvector_trick_that/fdgej4q?utm_medium=android_app&utm_source=share

Am I right in this comment

[dusky epoch]

honestly i kinda don't like the word "transformation" due to the amount of confusion it can cause sometimes

[alpine echo]

Man... But it's sooo... Pleasing.. haha

I was literally in shock, when I first saw this interpretation

That question I sent, again confused me

Algebra's fine... But I don't know why, I never get a satisfaction learning that way :))

[dusky epoch]

alright wait ok

ARGH! I KNEW IT!

i fucked up my formula.

$C^{-1}AC = D$ can be rewritten as $A = CDC^{-1}$, not as $D = CAC^{-1}$!

[stoic python]

Ann:

[alpine echo]

Yes precisely

That's my point

[dusky epoch]

christ, what a brainfart i had there

yeah alright so

so i can attempt to explain this from a transformation point of view

[alpine echo]

So when he found, $ A^{100}$ he actually did, $ C D^{100} C^{-1}$

Right ?

[dusky epoch]

last D should be C but yes

[stoic python]

Aravindh_Vasu:

[alpine echo]

But he never told that though?

[dusky epoch]

yeah he didn't mention that explicitly

i mean tbh not everything can be put in a geometric light as clear as this

and sometimes you gotta do some symbolic manipulations

like that's the entire point of linear algebra

to allow you to not rely only on direct visual intuition

[alpine echo]

Oh... But when you can use his tool to animate it yourself ... super fun

to allow you to not rely only on direct visual intuition

[dusky epoch]

good linear algebraic intuition isn't just about having a good geometric picture of what's going on, it's about being able to go back and forth between symbols and geometry as needed

[alpine echo]

Yes it's actually pretty tough to understand Abstract algebra

After having this visual aid

In 2D

Alrighty let me animate this though :))

Is there a channel, where I can post a video for checking, before I post it on YouTube ?

Checking for mathematical correctness

[dusky epoch]

you can put it on YouTube and make it unlisted

[alpine echo]

And then share it here itself?

[dusky epoch]

put the video link here

yes

[alpine echo]

Cool thanks. Wow you are so helpful. Seriously, thanks a lot.

Honestly, nobody's been this patient to me. Thanks again.

[sonic osprey]

It does mean that

[pliant harbor]

Quick proof: injective <--> dim Nul = 0 <--> rank = n <--> surjective. Also, dim Nul = 0 <---> 0 not an eigenvalue <---> det ≠ 0 <---> invertible.

[honest swift]

@pliant harbor did you catch my clarification last night about your question?

[thorn prairie]

Im having linear algebra on uni this semester. Is it an easy subject and how should i approach it?

[high zenith]

I am not entirely sure if this is the correct channel

but could someone explain to me what "skew-adjoint" means?

[dusky epoch]

uhhhhh

my best guess would be that an operator $A$ is called skew-adjoint if $A^* = -A$ but your book ought to have a defn

[stoic python]

Ann:

[pliant pollen]

Which of the following matrices may be obtained by performing exactly one elementary row operation on the matrix?
[
\begin{bmatrix}
1 & 3 & 0 \
0 & 1 & 3 \
0 & 0 & 1
\end{bmatrix}
]
A. [
\begin{bmatrix}
0 & 0 & 0 \
0 & 1 & 0 \
0 & 3 & 1
\end{bmatrix}
]
B. [
\begin{bmatrix}
0 & 0 & 1 \
0 & 1 & 3 \
1 & 3 & 0
\end{bmatrix}
]
C. [
\begin{bmatrix}
1 & 0 & 2 \
3 & 1 & 6 \
0 & 3 & 1
\end{bmatrix}
]
D. [
\begin{bmatrix}
0 & 1 & 0 \
0 & 3 & 1 \
1 & 0 & 3
\end{bmatrix}
]
E. [
\begin{bmatrix}
1 & 3 & 1 \
3 & 1 & 0 \
0 & 3 & 1
\end{bmatrix}
]

[stoic python]

OPz qt:

[pliant pollen]

ik for sure b

ah formatting messed up

Which of the following matrices may be obtained by performing exactly one elementary row operation on the matrix?
[
\begin{bmatrix}
1 & 3 & 0 \
0 & 1 & 3 \
0 & 0 & 1
\end{bmatrix}
]
A. [
\begin{bmatrix}
0 & 0 & 0 \
0 & 1 & 3 \
0 & 0 & 1
\end{bmatrix}
]
B. [
\begin{bmatrix}
0 & 0 & 1 \
0 & 1 & 3 \
1 & 3 & 0
\end{bmatrix}
]
C. [
\begin{bmatrix}
1 & 3 & 0 \
0 & 1 & 3 \
2 & 6 & 1
\end{bmatrix}
]
D. [
\begin{bmatrix}
0 & 0 & 1 \
1 & 3 & 0 \
0 & 1 & 3
\end{bmatrix}
]
E. [
\begin{bmatrix}
1 & 3 & 0 \
3 & 1 & 3 \
1 & 0 & 1
\end{bmatrix}
]

[stoic python]

OPz qt:

[pliant pollen]

so I guess my question is, ik b is an answer d, can be done if one elementary row operation is applied twice, but there has to be another matrix

<@&286206848099549185>

[tardy dawn]

@pliant pollen tell me, what are the elementary row operations?

[pliant pollen]

swapping two rows

adding two rows

multiplying a row by a constant

[tardy dawn]

correct. so you said you know B is an answer. what ERO did you use to get B?

[pliant pollen]

swap row 1 and 3?

[tardy dawn]

very good!

now, what's the ERO that you'd have to apply twice to get D?

[pliant pollen]

swap 1 and 3 and then 2 with new 3

[tardy dawn]

right. now because that's more than exactly one elementary row operation, I would personally rule out D.

there may be, however, one more besides B

[pliant pollen]

A is out

[tardy dawn]

the hint is that it depends on your definition of elementary row operation. I learned that there are two EROs:

1. swap two rows
2. add a constant multiple of one row to another row

if you accept my definition of ERO, then there's another that you can do with exactly one ERO. but it can't be done with your definition of ERO.

[pliant pollen]

it's probably your definition I need then

[tardy dawn]

ok. so which is the other one that you can get with exactly one ERO (my defn)?

(look carefully, it's there.)

[pliant pollen]

ah c

[dusky epoch]

was row scaling not an ERO for you or did you have to like. take precautions against adding -1 times a row to itself @tardy dawn

[tardy dawn]

and what did you do to get that one?

[pliant pollen]

2 * R1 + R3

[tardy dawn]

bingo! I'm proud of you, you got it.

[pliant pollen]

thank u sm

[tardy dawn]

@dusky epoch I mean, yeah, you couldn't zero out a row on purpose. I omitted that in my definition above.

[quasi vale]

Hello, just started vector spaces, subspaces, spanning, etc. Have a question

$\mathbb R^{2} is not a subspace of \mathbb R^{3}$

Why is this

[stoic python]

Sup?:

[cursive narwhal]

Uh, is that what your textbook said?

[quasi vale]

wdym?

No, the teacher told us to go read about subspaces. Was looking at a video and it said R^2 is not a subspace of R^3.

It also mentioned the reasoning, but I didn't quite understand.

[dusky epoch]

R^2 and R^3 are literally disjoint

they're completely different spaces

and no vector is in both at once

[quasi vale]

Ah.

[dusky epoch]

not even the zero vector of either space

[rancid ore]

im doing practice questions for an exam next week

and im stuck on a question

[dusky epoch]

okay

post it

[dusky epoch]

woah

uh

[rancid ore]

i have highlighted the question in blue

[dusky epoch]

please crop next time

[rancid ore]

sorry ahahah

[dusky epoch]

this is a pain to view on mobile

[rancid ore]

lemme take a pic of my page to make things simplet

[gray dust]

https://i.gyazo.com/ad16410f886d36ddd743ad5825d7a887.png

[dusky epoch]

okay... so where exactly are you stuck @rancid ore

[rancid ore]

i feel i have a decent understanding of what i need to do

[dusky epoch]

then do it

[rancid ore]

create an upper triangular matrix

using the basic rules

then solve via back substitution

[dusky epoch]

do it. and then come back and show us where you get stuck.

[rancid ore]

where i struggle

is getting the upper triangular

we were told to avoid zeros on the diagonal

[dusky epoch]

do it as far as you can. then come back with your work and show us where you got stuck.

[rancid ore]

ive got like 2 pages of trying different things

i can get it down to one variable away

[dusky epoch]

show at least one attempt then

so that i can see if it's salvageable

[rancid ore]

okay give me a minute

btw

i already know the expected answer as ive solved the equations via a different method for question 3

[dusky epoch]

okay

[rancid ore]

I

this is one way

but the answer was wrong

[dusky epoch]

uh

[dusky epoch]

okay so like

[rancid ore]

then theres some more tinkering

[dusky epoch]

wait

[rancid ore]

i feel like im missing something obvious

i tried to keep my steps clear

sorry about the handwriting

[dusky epoch]

found your mistake in the first attempt

"add 2 times the third equation to the second"

you incorrectly calculated 5/2 + 2 * (-2) as 1/2

rather than -3/2 as it should be

but in any case

you went an inefficient and error-prone route

bc you introduced fractions

[rancid ore]

yeh i feel like theres maybe a row i could multiply by a constant to make it simpler

[dusky epoch]

those are easier to screw up on

you could subtract twice the second row from the first

then switch the first two rows

[rancid ore]

then subtract the the new second row from third?

wait no

i know once we get to this stage

i should only use row two and three

[dusky epoch]

yes

[rancid ore]

so could i multiply row 2 by two and row three by 5

then subtract 2 from 3

[dusky epoch]

you fucked up

er

wait

[rancid ore]

bruh

[dusky epoch]

hang on

[rancid ore]

1-6 right?

[dusky epoch]

yeah your second row should be ( 0, -5, +2 | 6 )

[rancid ore]

ah shit

okay

wait

2 x -1

ah minus

[dusky epoch]

you're subtracting 2*(-1).

[rancid ore]

okay

[dusky epoch]

now you can subtract twice the third row from the second

[rancid ore]

duhhhh

umm okay

gimme a min

theb subtract 2x second from third?

[dusky epoch]

well your second row should have become ( 0, 1, 0 | 0 )

you can just zero out the entire second column (except for that 1) in one fell swoop

[rancid ore]

wait

-(2*-2)

so its essentially -5+4

[dusky epoch]

yes so you get (0, -1, 0 | 0)

[rancid ore]

ahh

you wrote positive one before

i thought i messed up again

[dusky epoch]

which can be transformed into (0, 1, 0 | 0)

by a multiplication by -1

[rancid ore]

yeh

which means y -0 which is correct

y=o

0

[dusky epoch]

y=0 yes

[rancid ore]

got it thanks

its the very first bit i struggled with most

when you didn 2x second from first

i just couldnt see it

[dusky epoch]

you have a 1 in the second row

1s are good to have

[rancid ore]

thanks

illl keep that in mind

ive got one more to do

ill see how i get on

[rancid ore]

@dusky epoch

[dusky epoch]

h

[rancid ore]

i seem to either get stuck with one variable too many

[dusky epoch]

can this be the last one

[rancid ore]

or get the upper triangular with wrong answer

yes this is the last one

ill send my opages

ive tried a bunch

[wintry steppe]

How do you get the sin portion of the combined function

[pliant harbor]

This doesn't look like linear algebra.

[south wadi]

1 -4 1
2 -1 -3
-1 -3 4

im kind of confused for this one on setting up the pattern

1 0 0
0 1 0
0 0 1

because this one only has an x1 and x2

ususally i deal with x1 , x2 , x3 and solve trying to get the pattern i posted above

Help pleas

e

ping me

[magic slate]

@south wadi

[undone crag]

Can someone help me with this question? I tried solving the systems but got nowhere

[magic slate]

@undone crag

[undone crag]

thanks @magic slate but we have only done up to the dot product so i dont think my prof will accept gauss-jordan elimination for this homework

[magic slate]

hmm

really?

that was the first thing i covered

but i can try to redo it

well

i dont see how it could be any easier than using gaussian elimination

if he wouldnt accept that could you just rewrite it as a regular system of 2 equations and try some elementary techniques

its equivalent to what i did

but with equations instead of a matrix

[limpid dagger]

@magic slate your name is a word play on the euler mascheroni constant, right?

[magic slate]

yeah

its from flammable maths

just wanted to rep the flammily

[limpid dagger]

nice

my friend, just today, said it like that

[magic slate]

lol, must be a fellow flammable maths fan

[limpid dagger]

no, I don't think so, this was his first time hearing about it

[magic slate]

oh ok

[dreamy cedar]

Flammy maths

[pliant harbor]

I oughta get hipster points for having come up with Oily Macaroni 3 years ago.

Also, the question said "Find a,b,c", not "Find a,b,c and tell us how you got them or we'll assume you used blackmail." So you can use whatever method you want, then write down the values that work and verify that they work.

[dusky epoch]

:?

[urban spear]

it's kind of implied that you should show your working when answering maths problems, is it not @pliant harbor ?

[magic slate]

but the guy was saying that he couldnt use gaussian elimination

which i dont think makes sense

but who knows

[wintry steppe]

I'm confused. I thought a_i is a column vector, wich should be in R^m given that A is in R^(m x n)

[empty copper]

There is no mistake

A is an m by n matrix, which means that it has a total of m rows, each of length n

That's why the a_i, which represents one such a row, is of length n

[slender shale]

hi can someone explain pages 35 to 39 to me

https://www.math.ucdavis.edu/~linear/linear.pdf

[empty copper]

If you've read them yourself, please indicate which parts exactly you have trouble understanding

[dusky epoch]

gonna take a look anyway, but we're still gonna need that info

[slender shale]

like i understand how MX=V, and i get the definitions on page 37 and 38, but the part thats confusing me is how they got that MY1=0 says the subseqent vector equation solves the homogeneous system

[dusky epoch]

is it this paragraph which confuses you?

[slender shale]

no the next page is what confuses me

the part after how they solve the original system of equations

[dusky epoch]

the example?

[slender shale]

yes

[dusky epoch]

this?

[slender shale]

yes

[dusky epoch]

the homogeneous system in matrix form is expressed as MX = 0

by defn

[slender shale]

but how do they get that those values for the unknowns solve the homogeneous system

[dusky epoch]

MY_1 = 0

if you replace X with Y_1 in MX = 0, you get a true statement

don't overthink

[slender shale]

ah shit i got it

damn fuck vectors fuck matrices

idk why they have to be so damn important

[dusky epoch]

,,,

i don't really think that's a healthy stance to have towards any kind of math

[slender shale]

i love math of all sorts, but i have a love/hate relationship with this particular class

the challenge is fun but sometimes it makes me want to rip hairs out

[dusky epoch]

that's just how it be sometimes

[slender shale]

first attempt i got a D. now im redoing it on my own time to understand it

because i want to do ML for a career

i remember my first test in LA i got a 63. it was my lowest grade in math since 3rd grade

[dusky epoch]

honestly i gotta say like... i don't think there's ANYONE out there who hasn't at some point wanted to rip their hair out bc of something they found difficult in math

[slender shale]

idek how to visualize these vector and matrix operations. like what does it physically or geometrically mean to multiply a matrix by a vector and so on

i know how to visualize a derivative for example. it gives the slope of the tangent line

[compact light]

3blue1brown has a series in which he visually explains lineal algebra

was incredibly usefull to me

[slender shale]

i'll have to check that out

[compact light]

https://gyazo.com/0328bdc1c2b4e69cfca58c68b0c35d48

how could i diagonalize this?

Im stuck at getting the eigenvalues

when doing the determinant of nuc(A-kI) i get
(a-k)^2*(b-k)-2(a-k) = 0

and i dont know how to simplify it to get the k

[dusky epoch]

well you can factor out (a-k)

[compact light]

I've come this far

but i cant get the k with the 2n grade equation, im stuck there

[lone quail]

I know that given three points A,B and C the equation of the plane contained by all three points is given by the determinant of the above matrix

My question is, why is the top row not simply x y z

As the two other rows are basically direction vectors

[lone quail]

<@&286206848099549185>

[quasi vale]

Hi. I'm asked to check if three vectors in R^4 are linearly independent or dependent.

Vectors are: <1,3,-1,4>, <3,8,-5,7>, <2,9,4,23>. What I did was C1V1+C2V2+C3V3=0, where c1,c2,c3 are constants.

I made a matrix, tried to solve it by echolon form and I'm getting two equations: C1+3C2+2C3=0, 3C3-C2=0.

If c1,c2,c3 are all 0, then the vectors are linearly independent. What I did was take some value of c3 and got back some values of c2 and c1. This shows that not all constants are zero and the equations are being satisfied, therefore the vectors are linearly dependent. Is my working right?

[slow scroll]

@lone quail Im pretty sure by setting the determinant equal to zero, you are looking for the values of x,y,z such that the first row lies in the plane of the other two rows

[lone quail]

Oh alright thanks i think i got it

[gleaming topaz]

@quasi vale Thats the way to go but I got another answer by calculating it online

[quasi vale]

Wdym?

Is there a better method?

They are linearly independent?

[gleaming topaz]

No, they're linearly dependent, but you've how they dependent wrong. It should be https://gyazo.com/f6aefa07391cfeb71dbbe456d774cb6a

Your second row is right

But something is wrong in the first

[quasi vale]

What I did was choose c1=1

sec

My bad

[gleaming topaz]

What I mean is you're supposed to get that C1+11C3=0

[quasi vale]

Hm

[gleaming topaz]

Well, when I see your issue

It's not in RREF

[quasi vale]

Wdym?

[gleaming topaz]

Your first row is not supposed to have any C2's

[slow scroll]

It’s enough to check that the kernel of the vectors written as the columns of a matrix is trivial. I.e. check that the rows/columns of the REF or RREF are linearly independent

[gleaming topaz]

I guess yeah my bad shouldn't have focused on that if the focus was to check linear independence

[quasi vale]

Yeah it's not in REF or RREF my bad

[slow scroll]

By the time you write it in REF you should just be able to “look at it”

[quasi vale]

actually, it is in REF

So I should solve by RREF?

[slow scroll]

It doesn’t matter...

[quasi vale]

What exactly did I do wrong? My row operations are correct

[slow scroll]

Nothing? Are the rows of the REF matrix linearly independent?

[quasi vale]

Yes, they are linearly independent in the REF matrix.

The first two rows.

[slow scroll]

Ok, then that’s all. The vectors are linearly independent. This all comes from the fact that row operations preserve dimension of fundamental spaces. I.e. if the original vectors are linearly independent then the REF of the column vector matrix will be linearly independent as well

Wait a second

[quasi vale]

But my answer is coming as linearly dependent.

And the book also says that.

[slow scroll]

Oh you have a 0 row?

[quasi vale]

Yes, two 0 rows.

and the vectors are 4 dimensional

idk how to code or I would show my matrix

[slow scroll]

Did you write the vectors as the rows of a matrix?

[quasi vale]

Yes.

[slow scroll]

Yea I don’t think you can do that

[quasi vale]

No, no

Actually column

[gleaming topaz]

Yeah he got it right, his vectors are linearly dependent

[quasi vale]

I wrote the vectors as columns of the matrix

[gleaming topaz]

By turning it into rref you can see that x1=-11x3, x2=3x3 and x3 is a free variable therefore the trivial solution is not the only one to your system of equations

[quasi vale]

By trivial, you mean x1,x2,x3 = 0?

[gleaming topaz]

Yes, if the case was that the only solution to your system of equations is x1=x2=x3=0 when you have it in RREF then they're linearly independent

but now you see that x3 can be any real number and x1=-11x3 and x2=3x3

so they're linearly dependent

[quasi vale]

Alright thanks. And I can also check in REF right? Or RREF is necessary?

[gleaming topaz]

REF works

[quasi vale]

thanks.

[quasi vale]

Let V=P_3 be the vector space of all polynomials of degree <= 3 over R. Determine whether u, v, w belonging to V are linearly independent or linearly dependent.

U=x^3 - 4x^2 + 2x + 3, V= x^3 + 2x^2 + 4x - 1, W= 2x^3 - x^2 -3x +3

I'm not sure how to write this in matrix form, need some help

[feral grove]

well think about what adding polynomials means, and see if there's a way to rewrite them equivalently as vectors in R^4

[quasi vale]

How'd you know we have to write them as vectors in R^4?

[feral grove]

because each polynomial has 4 terms

[quasi vale]

And we can think of each term as the individual component of the vector?

Sorry if I asked a stupid question. <@&286206848099549185>

[feral grove]

oh

[quasi vale]

Hey.

[feral grove]

but yes, we can think of coefficients of the polynomial as entries in a vector

mb i got distracted

[quasi vale]

np[

[feral grove]

so the idea is adding polynomials amounts to adding like terms

[quasi vale]

Alright, so I can write these 3 vectors as vectors of 4 dimensions

[feral grove]

so x^2+1+2x^2+2=3x^2+3, and you can notice that this works the same way as adding vectors (1 1)^T and (2 2)^T

so yes you can write them as vectors and then do all your known operations

[quasi vale]

wdym by (1 1)^T

or (2 2)^T

[feral grove]

oh i meant

(1 0 1)^T and (2 0 2)^T

those are vectors

T means transpose as in we flip it along a diagonal so it's actually a column vector

i'm just too lazy to latex it so i'm writing the vectors which are in R^3 in text

[quasi vale]

It's necessary to write vectors in column in a matrix?

[feral grove]

you could write them in rows but then you'd have to perform column operations

[quasi vale]

Oh.

So, U = x^3 - 4x^2 + 2x + 3, how do you propose I write this as a vector?

[feral grove]

( 1 -4 2 3)^T

[quasi vale]

Got it

Thanks man

[feral grove]

np

[quasi vale]

Wait.

Just one more question.

How is it that we apply row operations when we have vectors as columns and column operations when the vectors are written as rows?

[feral grove]

so if we say is a matrix A=(v_1 v_2 v_3) where v_1,v_2,v_3 are vectors in R^3, if we take A^T rows become columns and vice versa, so any row operation on our column vector becomes a column operation on our row vector by transposition

[quasi vale]

Im not sure I follow, I'll have to watch a video or something

But thanks bro

[vital fjord]

Right some on part b

Let our general vector be $p=ax^2+bx+c$ however is that equal to $p=\begin{bmatrix} a \ b \ c \end{bmatrix}$ or $p=\begin{bmatrix} c \ b \ a \end{bmatrix}$ ?

[stoic python]

Brandonhotdog:

[dusky epoch]

not only does that not make sense

the answer is still no

if you meant

${ax^2 + bx + c \mid a, b, c \in \bR}$

[stoic python]

Ann:

[dusky epoch]

as in, considering this set as a vector space

then it is distinct from $\bR^3$

[stoic python]

Ann:

[dusky epoch]

$ax^2 + bx + c$ is not a column vector; it is a polynomial.

[stoic python]

Ann:

[vital fjord]

Yeah, okay to be more specific, a vector space with, as said in the question, has the basis {1,x,x^2}

over the field of R

[dusky epoch]

IF you're considering this space equipped with the basis ${1, x, x^2}$, with the elements of it listed IN THAT ORDER,

THEN the COORDINATE VECTOR of $ax^2 + bx + c$ would be $\begin{bmatrix} c \ b \ a \end{bmatrix}$.

[stoic python]

Ann:

[vast torrent]

You need to pick an order, bases don't come with an order by default

a+bx+cx² ~ [c a b]^T is also an isomorphism

[vital fjord]

yeah I thought the column vector would have c at the top

[dusky epoch]

again

it depends on the order

[vital fjord]

okay, so let's set the scene

I'm in an exam

[dusky epoch]

it depends on what order you put your basis elements in

[vital fjord]

This question appears

that's all I'm given

[dusky epoch]

ok standard monomial basis

[vital fjord]

I ask the invigelator what order the basis is and he says go away

[dusky epoch]

so i suppose they're expecting you to use the order they list it in

[vast torrent]

sets don't have an order by default, you pick an order for these kinds of problemd

[vital fjord]

okay so in general for a basis ${x_1,x_2,\ldots,x_n} if P=\sum ^ n _{i=1} \alpha _i x_i = p=\begin{bmatrix} \alpha _1 \ \alpha _2 \ \alpha _3 \ \text{and so on} \end{bmatrix}$

[dusky epoch]

eugh bad tex

[vast torrent]

{1,x,x²}={1,x²,x}={x²,1,x}=...={x²,x,1}

[dusky epoch]

but yes once you fix an order the coordinates go top to bottom

[stoic python]

Brandonhotdog:

[vital fjord]

ok yeah may latex isn't the best but you see what I mean lol

and alright cool

thank you for the help

and I ask this because wouldn't it give me a different matrix for A depending on the way I order the basis?

[vast torrent]

Yes

It would

[quasi vale]

I have a question. Suppose we have a set S consisting of two 2 dimensional vectors and their span is R^2. Now if their span is R^2, that means they are linearly independent. But for the set S to be the basis for R^2, do we have to prove that the vectors indeed are linearly independent(by C1V1 + C2V2=0, where C1 and C2=0)or can we conclude their independency from the span?

[dusky epoch]

Now if their span is R^2, that means they are linearly independent.

do we have to prove that the vectors indeed are linearly independent

well if you assert that they are

then presumably you have some reason for it

[quasi vale]

Thing is, I'm watching a video on Khan Academy. We had two vectors and we proved their span is R^2. Now if their span is R^2, what Sal did was prove that they were linearly independent by using C1V1 + C2V2 = 0

But was that necessary?

[vast torrent]

@vital fjord the columns of A would be in a different order

[vital fjord]

yeah I thought so, so they'd have the same determinant

[vast torrent]

The sign of the determinant could change

[vital fjord]

I ended up getting $A= \begin{pmatrix} 2 & 1 & 0 \ 0 & 1 & 4 \ 0 & 0 & 0 \end{pmatrix}$

[stoic python]

Brandonhotdog:

[vital fjord]

luckily like all great problem sheets (/s) they didn't give us an answer sheet so I can only hope it's right lol

[vast torrent]

If the columns correspond to D(1),D(t),D(t²) through the appropriate isomorphism to R³ then its correct /shrug

[vital fjord]

ay nice lol

Also am I right to do $ \begin{pmatrix} 2 & 1 & 0 \ 0 & 1 & 4 \ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} a \ b \ c \end{pmatrix} = \mathbf{0}$

[stoic python]

Brandonhotdog:

[vast torrent]

But there's nothing God-given consequence-of-universal-laws correct order to {1,t,t²} since sets don't have an order

[vital fjord]

and then $P= \begin{pmatrix} a \ 2a \ \frac{-1}{2}a \end{pmatrix}$

[stoic python]

Brandonhotdog:

[vast torrent]

The most common one is to pick the order (1,t,t²),notice i change from curly brackets to round brackets

Round brackets imply an order

[vital fjord]

oh, that cool, and tbh I feel like it's pretty standard for test just to implicitly leave out so much needed info

[vast torrent]

Uh I'd have to work out what the kernel is, let me see if i can do it in my head

[vital fjord]

like for example, they never mention over what field their vectorspace in which could affect it, obviously we assume R

[vast torrent]

2a=-b, b=-4c

2a=4c

Your signs are off

[vital fjord]

yeah, just noticed lol

but the method of doing simultaneous equ to find kernel is the right way to go about it?

[vast torrent]

If the matrix is already in row echelon form then that's generally the easiest

You can go to reduced row echelon form first if you want

[vital fjord]

but if they just threw some random matrix at us

[vast torrent]

I'm calling it row echelon even though the first pivot is 2, some books need all pivots one to be called row echelon

[vital fjord]

so you reckon in a test that'd be best?

[vast torrent]

I personally find it easiest usually to go to row echelon and then convert to equations. It's 100% valid to go to reduced row echelon first if that's easier for you

[vital fjord]

okay cool, thanks for the advice

[vast torrent]

Really you can try solving the equations in the original form without row operations

Whatever is easier given the context

The field is generally R unless you need it to be C for a problem to be solvable

Lots of problems about matrices of real numbers don't have a solution unless you expand the field to C and consider R as a subset of C in the usual way x <-> x+0i

In particular problems involving finding a change of basis matrix so that A is diagonal or triangular

Those problems are generally only solvable with complex numbers as the field

[vital fjord]

ayy, yeah I need to learn diagonalizing matrices soon, it's on my test

[vast torrent]

Yeah so "diagonalize this matrix if possible" problems assume the field is C because it's often impossible otherwise

Sometimes it's impossible even over C and the best you can get is block diagonal

You'd still require C

[vital fjord]

a block diagonal?

https://blogs.sas.com/content/iml/files/2012/11/blockmatrix.png

oh, that's pretty odd but cool

[vast torrent]

Yeah, even better actually every matrix is equivalent to a block diagonal triangular matrix with 1s or 0s on the off diagonal

So the one in your picture is worse in that it isn't triangular

This is called the jordan normal form and every matrix is equivalent to one

,w jordan normal form {{2,1,0},{0,1,4},{0,0,0}}

[stoic python]

Display results online and refine query

Requested by gfauxpas#9497

[vast torrent]

You got lucky here, the jordan normal form is diagonal and you only needed real numbers

[vital fjord]

,w jordan normal form {{3,1,-5},{6,-4,2},{0,1,4}}

[stoic python]

Display results online and refine query

Requested by Brandonhotdog#1111

[vital fjord]

scary

pretty cool though

Also by "qualitative structure" reckon I could just say "The matrix transforms vectors in R^3 to R^2"

or "The solutions of this will be vectors in R^2"

hmm, I think what I just said doesn't make sense lol

"The solutions are the span of 2 linearly independent vectors in R^3"?

[vast torrent]

I'm not sure if "qualitative structure" means something specific or if they just want your conclusions on it

[vital fjord]

Back at it again

So I feel like $det(v_1,\ldots,v_{n-1},e_i)$ is like some weird generalised dot product between w and e_i

[stoic python]

Brandonhotdog:
Compile Error! Click the reaction for details. (You may edit your message)

[dusky epoch]

cross

well

ok no

nvm

ignore me

[vital fjord]

do you know what that sort of thing is called cause I could barely find anything online that had a determinant of a bunch of vectors and info about it

like I know you can write a determinant like that

but like beyond that I don't know what it's good for or how it can be used

[vast torrent]

@vital fjord I think I figured it out

recall that the orthocompllement of the image of a matrix is related to its transpose in the following way

$(\Im(A))^\bot = \ker(A^\top)$

[stoic python]

gfauxpas:

[vast torrent]

that means that to show w is perpendicular to to v1,v2, ..., v_{n-1}, it is sufficient to show

$\begin{bmatrix} - & v_1^\top & - \ - & v_2^\top & - \ & \vdots & \ - & v_{n-1}^\top & - \end{bmatrix} w = 0$

[stoic python]

gfauxpas:

[vital fjord]

what is \bot ?

[vast torrent]

orthogonal

perpendicular space

this should make sense because if vectors are perpendicular

but $v \cdot w = v^\top w = 0$

[stoic python]

gfauxpas:

[vast torrent]

so the only way that matrix on w can be zero is if each row v^T is orthogonal to w

[quasi vale]

Lipschitz, can you help me with equations of subspaces?

[vast torrent]

no, I have to go actually, i'll be back in like half an hour though. probably.

good luck

[quasi vale]

np

[vital fjord]

hmm, I'll have to look into perpendicular spaces but thanks for putting me in the right direction @vast torrent

[vast torrent]

@vital fjord actually what I said isnt enough on its own 😦

still thinking on it

@ me if you figure it out

[elder creek]

Anyone have a good book for learning finite vector spaces? Preferably in an eli5 manner

[pliant harbor]

@honest swift I read and understood the definition of f_a, but I'm not sure how this will help me prove or disprove f=0.

[honest swift]

@pliant harbor
There is such an f_a for every element a of the Hamel basis, which consists of your original set B = {(1,0,0,...), (0,1,0,...), ...}) as well as an uncountable number of other sequences. For one such of these other sequences (call it a), the function f_a is then a nonzero functional that vanishes on every vector of B.

[vital fjord]

@elder creek Have you tried Mathematical Methods for Physics and Engineering by M. P. Hobson? It has lots on linear algebra and other important topics in general

I don't know specifically how much it has on finite vector spaces but I'd stay say the book's worth looking into if you haven't already

[slow scroll]

does he mean finite dimensional vector spaces or vector spaces over finite fields?

[elder creek]

Vector spaces over finite fields

But no I have not. I just finished calculus and am venturing into linear. Will definitely read. Thanks!

[wintry steppe]

Are the integers modulo (prime squared) a field?

It is a semiring but not sure about field

[quartz compass]

I assume you already know the integers modulo a prime is a field already

so try to focus on what changes between the two and run through the list of axioms

[wintry steppe]

yeah so I think its a field as well

checked the axioms

[quartz compass]

check them harder

something about going from a prime to a prime squared changes something

[pliant harbor]

@honest swift How do you know that a Hamel basis exists? Are we using the fact that every vector space has a basis?

[honest swift]

Yep. (And so the axiom of choice is used, am not sure if you can prove it without.)

[pliant harbor]

Just one small problem. Z^w is not a vector space, so how do we know a basis still exists?

I was convinced it's a vector space, which would wrap up the problem for good, but then I remembered the first axiom of a vector space: "Let F be a field..."

[honest swift]

Oh I totally missed that this was over Z^w not R^w lol. Will have to have another think,

Yeah in fact that module is known to not be free abelian, so that argument does not work.

[shut sandal]

Please do correct me if I'm wrong (linear algebra):

Turning a matrix A into row echelon form (not reduced), shows that:
1. The number of nonzero rows is the rank(A) and also the dim of the row space. Finding a basis for the row space would require finding which of those nonzero rows are linarly independent (or are they all linearly independent?)
2. The pivot columns indicate which respective columns of the original matrix A compose the basis of the column space. The number of pivot columns is the dim of the column space.
3. Turning the matrix Aᵀ into row echelon form (not reduced) would indicate the opposite things:
The rank(Aᵀ) is the dim of the column space, and the pivot columns would indicate which respective rows of the original matrix A compose the basis of the row space.

So, summing up, to find the row space I need to either turn A into ref, and check if the nonzero rows are linearly independent, or turn Aᵀ into ref, and look at the pivot columns, the respective rows of which in A would be the basis of the row space.
To find the column space I need to turn A into ref, and look at the pivot columns, or turn Aᵀ into ref and check which of its nonzero rows are linearly independent.

It is also true that
dim(row space) = dim(col space) = rank(A) = rank(Aᵀ) = nonzero rows in row echelon form of A or Aᵀ

[silver ore]

I have no idea that ii even is

I kinda feel lost when I read it

and in general, I am really unsure about double dual spaces

and how to think about them

please help

Does double dual V** only make sense if there is a element from v that it acts on?

like say a in V* and ev in V**

what is ev(a)?

V** are mappings that sends stuff in V* to a field right?

so is ev(a) a field?

[hollow phoenix]

ev(a) = a(v)

[silver ore]

or is it only in a field if say for v in V ev(a)(v)=a(v)?

[hollow phoenix]

ev(a) = a(v) which is an element of the field

[silver ore]

right, but where did that v come from

[hollow phoenix]

Wait what is ev

[silver ore]

its the evaluation map

[hollow phoenix]

Do u mean e_v?

Oh

Well evaluation map needs a point

[silver ore]

so ev(a)(v)=a(v)

[hollow phoenix]

Yeah that’s right

You would write it as ev_v (a) rather

I mean I prefer that

[silver ore]

oh ok

[hollow phoenix]

Where did it come from?

It’s the definition of ev_v haha

[silver ore]

I still need help with 1(a) ii

[hollow phoenix]

Err

I don’t get where u got the evaluation map from

I mean it doesn’t come up in tgis partgicular question

[silver ore]

it came from my notes

[hollow phoenix]

Oh ok

It’s not relevant for this question

Since u wanna prove it’s an isomorphism

Why not try proving surjectivity first

[silver ore]

Ah, I read the notes wrong

anyways

they put in ev(v)(a) instead of ev(a)(v)

and I read that

[hollow phoenix]

You know a linear map is determined by its values on a basis

So take som arbitrary F in V^I. Suppose F(i) = v_i for each i, where v_i are arbitrary

The corresponding linear map in L(F_I, V) would be

The map F’ that sends e_i to v_i

Then indeed theta(F’)(i) = F(e_i) = e_i as required

Injectivity can be proven similarly.. lemme know if you have troubles

[silver ore]

I'll give it a go

[lone quail]

Why is the first set of equations equal to the bottom one in parametric form?

Whete is the y=tz coming from

[lone quail]

<@&286206848099549185>

[quasi vale]

Find an equation or equations of the subspace W of R^3 spanned by: <1,-3,2> and <-2,0,3>. Need some help here getting the equation. I was able to do the previous part where it was an equation of line.

[lone quail]

Nvm i got it it was a printing error

You can use this matrix:

Expand the determinant and you get the equation of the plane spanned by the two vectors

[dusky epoch]

[quasi vale]

Thanks, it worked. But can anyone explain why we solved for the determinant?

[steady fiber]

because the equation of a plane can be found using the normal of the plane, the normal of the plane is just the cross product of two tangent vectors to the plane, and cross product can be represented using a determinant of a matrix (although I dislike doing cross product that way)

[quasi vale]

Ah.

We put determinant = 0, can you also explain that please?

[digital hazel]

I'm not sure, but my guess is because the determinant is the volume of parallelpiped formed by the three vector. Since the span is a plane and the volume of a plane is 0. Therefore, det = 0??? @quasi vale

[quasi vale]

Hm, could be. Thanks for replying

[quasi vale]

If anyone else can provide some insight to why the determinant is 0 when solving for the plane spanned by some vectors? <@&286206848099549185>

[feral grove]

you're gonna have to be a little more specific, so like 3 vectors that span a plane in R^3, and we take the determinant and we get 0?

[quasi vale]

I'll show the question.

Find an equation or equations of the subspace W of R^3 spanned by: <1,-3,2> and <-2,0,3>. Need some help here getting the equation. I was able to do the previous part where it was an equation of line.

Was able to do it by the determinant method. But why is it equal to 0?

[feral grove]

tbh i'm not familiar with this technique

[quasi vale]

No problem.

[quartz compass]

I think I see what you're doing

if you take the determinant of 3 vectors, it gives you the signed volume of a parallelepiped made from it

so if you put in those two vectors with <x,y,z> and set the determinant equal to 0, you're basically forcing the volume of this parallelpiped to be 0

which would force the vector <x,y,z> to always be smashed down to the plane spanned by those other two vectors

if you have any questions just ask @quasi vale

[quasi vale]

signed volume?

Ah I get it now

But what did you mean by signed volume?

thanks @quartz compass

[quartz compass]

signed volume just means it can be negative or positive depending on the orientation

if you flip the order of vectors in your determinant, it flips the sign

have you learned the cross product and right hand rule?

it's really the same thing as that

[quasi vale]

I know that the sign flips but no I haven't learned about cross product and right hand rule

But thanks bro, at least now I understand what we're doing.

Thanks\

[quartz compass]

you're welcome

[lone quail]

If

[restive shuttle]

hi i need help!

so for part a, I got:

x2 = 1

x4 = 10

x1 + x2 + x3 = 2(x4 + x5)

x2 + x3 + x4 = 2(x5 + x6)

as my set up before making the augmented matrix

but someone on chegg got:

x1 + x2 + x3 = (1/2)(x4 + x5)

x2 + x3 + x4 = (1/2)(x5 + x6)

for the 3rd and 4th equations

so my question is: What does "Any 3 adjacent tasks will take half as long as the next two tasks?", is my interpretation correct, or is the guy on chegg's interpretation correct?

<@&286206848099549185>

[gray dust]

use the helpers tag only if no one responds after 15min

[restive shuttle]

oh ok sorry!

[gray dust]

so you're using x not t to denote task time

[restive shuttle]

yes

[gray dust]

so yea your interpretation of the above statement is off

x1+x2+x3=total time for 1st 3 tasks

"Any 3 adjacent tasks will take half as long as the next two tasks" means doing tasks 1 to 3 should take half the time to do tasks 4+5

[restive shuttle]

OH

thank you!

[gray dust]

yup np

[pliant bison]

hey

im trying to solve this, would appreciate if someone can help

1 2 -3 6
2 -1 +4 1
1 -1 1 3

solving this matrices (redused row echleon form)

I can show you what I have done

last row should be -9/5, not -26/5 (unless i did it wrong again)

<@&286206848099549185>

nvm looks right

got the right answer xd

[wintry steppe]

I want to learn how to find the range of a matrix. I read on my math course: "the range of a matrix is the maximal number of vectors (lines or columns) that are linearly independants" what does the course mean by linearly independents?

[dusky epoch]

the range... do you mean the rank?

[wintry steppe]

probably my course is not in english. I mean the rg(A)

[dusky epoch]

le rang

oui c'est "rank" en anglais

l'indépendance linéaire est un concept très basique et très important, si tu sais pas ce que c'est il te faut vraiment lire ton livre d'algèbre linéaire

parce que tout livre en donne la définition

et si c'est un BON livre y aura plein d'exercices sur le sujet

[wintry steppe]

rang en français = rank en anglais?

[dusky epoch]

oui

me demande pas pourquoi c'est ça, je suis pas une linguiste

[wintry steppe]

Je ne suis pas en train de lire un livre. Cette fois ci c'est un vrai cours de math. Je ne suis néanmoins pas fermé à l'idée d'en lire un.

Dans mon cours on lit juste la définition de rang:

"Lerangd'unematriceA,notérg(A)estlenombremaximaldevecteurslignes(oucolonnes)linéairementindép endants" ils n'en disent pas plus :p

je vais comme même vérifier si je n'aurai pas oublié une définition qu'ils auraient mit plus tôt

test de connexion

dans mes cours j'ai :
Cours 1 - Eléments de logique
Cours 2 : Techniques de preuve
Cours 3 : Relations et ordre
Cours 4 - Eléments d'arithmétique
Cours 5 - Résolution des systèmes linéaires

Est-ce qu'il y aurait un passage concernant les matrcies avec des lignes qui sont linéairement indépendantes en dehors du chapitre 5?

@dusky epoch

[dusky epoch]

euhhhhhhhhhhhhhhh

[wintry steppe]

Est-ce que tu me conseillerais de lire un livre d'algèbre linéaire en dehors des cours?

oui? ^^'

[dusky epoch]

où sont les espaces dans ce que t'as copié ptdr

d'ailleurs

euh

ça me semble bizarre

de donner une définition comme ça sans avoir défini l'indép-lin

[wintry steppe]

moi aussi. c'est pour çe que je demande.

je pense que j'ai compris

@dusky epoch linéairement indépendant = qui n'a pas de relation SUR LA LIGNE. C'est peut-être expliqué dans le chapitre Cours 3 : Relations et ordre

[dusky epoch]

nnnnnnnnnnon non non non non pas du tout

ok bien

voilà la définition

un ensemble de vecteurs ${v_1, v_2, \dots, v_n}$ est appelé \textbf{linéairement indépendant} si et seulement si $$\forall (c_1, c_2, \dots, c_n) \in \bR^n, \sum_{k=1}^n c_k v_k = 0 \implies (c_1, c_2, \dots, c_n) = (0, 0, \dots, 0)$$

[stoic python]

Ann:

[wintry steppe]

@dusky epoch c'est en effet étrange que l'on ne l'ai pas vu en cours. Je vais voir si ça correspond aux exemples du cours.

@dusky epoch what does the part at the sigma means? ck + vk = 0 or ck*vk = 0 or something else?

[dusky epoch]

oh jesus fuck you don't know sigma notation either?

[wintry steppe]

of course I know what is a sigma notation

[dusky epoch]

...

your question seemed to imply you didn't know what $\sum_{k=1}^n c_k v_k$ referred to

[stoic python]

Ann:

[wintry steppe]

start with k = 1. while k < n then ... do what is at the right

[dusky epoch]

...

no

[wintry steppe]

^^' I hope I am right

[dusky epoch]

no you're not

[wintry steppe]

😦

I may should learn more on proof and logic

[dusky epoch]

sigma notation specifically denotes SUMMATIONS

[wintry steppe]

ok I know then

it is what I need to know

[dusky epoch]

$\sum_{k=1}^n c_k v_k$ is short for $c_1v_1 + c_2v_2 + \cdots + c_nv_n$

[stoic python]

Ann:

[wintry steppe]

let me recap

ok

thank you

I understand

now

thank you 😦

[wintry steppe]

@dusky epoch I think I understand. can you give me a short exercise to see if I am able to determine the rank of a matrix in 2 minutes please?

[dusky epoch]

i can give you a matrix and ask you to determine its rank

if that's what you want

[wintry steppe]

yes I want it

[dusky epoch]

ok

$\begin{bmatrix}
3 & 1 & 4 & -1 & -5 \\
-9 & 2 & -6 & 5 & -3 \\
-5 & -8 & 9 & -7 & 9 \\
3 & -2 & -3 & 8 & -4 \\
-6 & 2 & 6 & -4 & 3
\end{bmatrix}$

[stoic python]

Ann:

[dusky epoch]

there

[hoary agate]

ogod

[wintry steppe]

@hoary agate what?

[hoary agate]

nothing, it's just a big matrix

[wintry steppe]

@dusky epoch let's go step b step. line 1: there is no 0 in the line. so I can know that at this steprg(A) is equals or superior to 1. I note that there are the number -1 and -5 but it should not cause trouble because these are relative numbers.

[dusky epoch]

i mean DUH of course its rank is at least 1

the only matrix with rank 0 is the zero matrix

which this matrix isn't

[wintry steppe]

@dusky epoch well I am lazy and I think the answer is the matrik is rank 5 because there is no line where there is a 0 and where there is at least one number different than 0

[dusky epoch]

it just so happens that your answer is correct

but your logic will fall flat on its face with this matrix: $$\begin{bmatrix}
3 & 0 & 4 & -1 & -5 \
-9 & 0 & 0 & 5 & -3 \
0 & 0 & 9 & 0 & 9 \
3 & -2 & -3 & 0 & -4 \
-6 & 0 & 6 & -4 & 3
\end{bmatrix}$$

[stoic python]

Ann:

[wintry steppe]

@dusky epoch in the same logic this matrix seems to be rank 1

[dusky epoch]

yeah now you're very, VERY wrong

[wintry steppe]

😦 ok I read the course again

[dusky epoch]

idk if your course will even help

get yourself a linear algebra book

it looks like your course thinks you know linear algebra already

[wintry steppe]

@dusky epoch does this book contain linear algebra https://www.amazon.fr/How-Prove-Structured-Daniel-Velleman/dp/1108439535/ref=pd_sbs_14_t_0/257-5252974-4212646?_encoding=UTF8&pd_rd_i=1108439535&pd_rd_r=fe927edc-59f5-4025-bc5a-b52a314408c2&pd_rd_w=IGMzD&pd_rd_wg=pxaGq&pf_rd_p=9b28d941-c13a-4c2b-b935-36854aa20020&pf_rd_r=04YGZ75PW1GPYZA2CJ1Z&psc=1&refRID=04YGZ75PW1GPYZA2CJ1Z ? do you suggest any book?

[dusky epoch]

no it doesn't

i suggest Linear Algebra Done Wrong

despite the name it is a decent book

[wintry steppe]

@dusky epoch cool the book is free and avaible online

[dusky epoch]

ugh

please don't ping me in every message, it's getting annoying

[wintry steppe]

ok

[uneven sonnet]

If i have a linear transformation from R^3 -> R^2 and the size of the basis of the kernel of the transformation is equal to 0, can I say that the basis of the image of the transformation is (1,0),(0,1)?

[wintry steppe]

Hello there
can you please make it clear for me how do we see the relation between the transpose of a matrix and the dual space?
what is the intuition behind the transpose ?

[dusky epoch]

If i have a linear transformation from R^3 -> R^2 and the size of the basis of the kernel of the transformation is equal to 0, can I say that the basis of the image of the transformation is (1,0),(0,1)?

you can say anything you want because that cannot happen

[wintry steppe]

yes, in this case the image is the whole R^2 space so yes!

[uneven sonnet]

How comes that cannot happen

oh yes you're right

I meant if the kernel of transformation is equal to 1

my bad haha

[wintry steppe]

@dusky epoch are you an engineer

[dusky epoch]

no

who the fuck do you take me for, rudy

[wintry steppe]

HAHAHAHA

LADW is an engineer book😕

LADR is a mathie book

:3

[dusky epoch]

look i'm not gonna suggest a book to an absolute rookie when it involves fucking full-on abstract algebra and unnecessary generality

[wintry steppe]

@dusky epoch do you consider that as "fucking full-on abstract algebra and unnecessary generalityط

؟

[dusky epoch]

LADR does things in a very general light bc it starts off with vector spaces over an abstract field rather than R or C

[wintry steppe]

LADR ??

[cursive narwhal]

Linear Algebra Done Right by Axler. I think that's what Ann is referring to.

[wintry steppe]

Thank you so much

@cursive narwhal yes, you’re correct

Linear Algebra Done Right is the book by Axler, and it’s known because Axler is very hesitant to use determinants.

[half ice]

LADW is not an engineer book lol. It's far too proof based for that

Honestly I don't suggest either book to someone new to lin alg

[pliant bison]

what book do you suggest @half ice

[stoic python]

more:

[vast torrent]

The main point of it is that

if $A$ is invertible, $A^{-1} = \frac{1}{\det (A)} \text{Adj}(A)$

[stoic python]

gfauxpas:

[vast torrent]

though this is hard to use by hand except in the 2x2 case

[stoic python]

gfauxpas:
Compile Error! Click the reaction for details. (You may edit your message)

gfauxpas:

gfauxpas:

[vast torrent]

I know at the very least that diagonal matrices with positive entries on the diagonal satisfy these hypotheses and the inverses of these matrix are also diagonal matrices and the entries on the diagonal are the reciprocal of the inverse

But I'm not sure what to do here. I tried to make an argument equating cofactors of A and A^-1 and arguing that the off-diagonal entries must be equal to their negative but that didn't really work

<@&286206848099549185>

[feral grove]

my class also used LADW

it's at the very least better than LADR

[rich hornet]

why the first blank has w ?

[dusky epoch]

what figure

[rich hornet]

w

w_1

where does it come from?

[tardy dawn]

read the entire problem carefully. the only other vector that's mentioned besides $\vec v$ is $\vec w$.

[stoic python]

ari5av:

[lone quail]

Book says that eigenvalues of matrix M are 1 and 2, but im getting a third order equation which cant be factorised with those eigenvalues

Is the book wrong? Could anyone check?

[dusky epoch]

1 is definitely an eigenvalue, i can tell you that much

[lone quail]

Because im getting this polynomial

If i put 1 in it doesnt make 0

[dusky epoch]

,rccw

[dusky epoch]

i am pretty sure that this is not the charpoly of your matrix

[lone quail]

What would it be? I checked 20 times and i cant seem to get where ive gone wrong

Ah nevermind

Found my mistake

Rip

Sometimes i fall so low xd

[wintry steppe]

where can I find the correction for each exercise of the book "linear algebra done wrong"?

[wintry steppe]

<@&286206848099549185> should I continue to ask or should I abandon?

[sonic osprey]

These are normally called errata, try searching for that

[wintry steppe]

@sonic osprey I found only typo corections

not exercises correction

[sonic osprey]

Then there isn't a list

[wintry steppe]

😦 ok

@sonic osprey did you read the book?

[sonic osprey]

No

It's possible that there's not really that many errors in the exercises to be corrected

[wintry steppe]

I am very disapointed. It is a big book

and somebody recommended me

this book

[sonic osprey]

why are you disappointed

[vast torrent]

I have a matrix A in GL(n,R) with all nonnegative entries and A-¹ also has all nonnegative entries

How many nonzero entries does A have, what are their locations, and what are the entries of A-¹ in terms of the nonnegative entries of A?

For starters I know that a diagonal matrix with +ve diagonal entries satisfies the hypotheses and

The inverse of a diagonal matrix has the diagonal of reciprocal entries

Other than that i tried to equate cofactor elements of A and A-¹ and figure out which entries are equal to their negatives

But I couldn't get that method to work

https://math.stackexchange.com/questions/3507738/invertible-matrix-with-nonnegative-entries posted on stack exchange

[wintry steppe]

what is echelon form

[dusky epoch]

http://lmgtfy.com/?q=what+is+echelon+form

[vast torrent]

<@&286206848099549185>

[wintry steppe]

@wintry steppe row echelon form is basically this

jus think of it as diagonal 1s

and beneath them are zeros

[dusky epoch]

@wintry steppe don't encourage them

[wintry steppe]

oh sorry

[lone quail]

I have to find the ortonormal base of these two vectors

(I think its called ortonormal in English but I’m not sure: its orthogonal and divided by its length)

The answer ends up with these two vectors (just separate them from matrix A)

I’m struggling in understanding how he came up with the second vector

[quartz compass]

it's not enough to just normalize both vectors

[lone quail]

They thought us to use gram Schmidt but I’m getting a different answer

[quartz compass]

you need to remove the projection of one vector from the other

[lone quail]

Yeah i did

I get something else tho

[quartz compass]

what do you get

GS should work

[lone quail]

Ah nvm i found my mistake

[quartz compass]

ok cool

[lone quail]

Actually

I did took away the project from v1

I should have taken it out from v2

Thanks

[quartz compass]

yup

you're welcome

[vast torrent]

orthogonal and unitized in English is "orthonormal" with a "th" @lone quail

[lone quail]

Kk thanks

[vast torrent]

[wintry steppe]

What did I do wrong

[gray dust]

https://i.gyazo.com/c96fff944f1cb4d6a5224630e49293c1.png

$13-3\neq-10$

[stoic python]

RokettoJanpu:

[sacred quest]

errrr my homework is asking to find the "corresponding homogenuous system" of an augmented matrix and I don't really know how to do that

that wasn't discussed in class at all

[gleaming topaz]

I think you're supposed to A|0 and find a basis for the null space?

[sacred quest]

care to elaborate?

Do you mean

Manipulate the equations to get a homogenuous system?

If I had the two rows (it's a 2x4 matrix) of the RREF form together than I would end up with a homogenuous equation

[ 1 1 -5 | 0 ]

like that

[gleaming topaz]

Could you post a picture?

[sacred quest]

aye

i did the general solution part, puzzled by the homogeneous aspect

[gleaming topaz]

I would assume that you replace 4 and 6 with 0's and get a basis for the nullspace?

Make sense?

[sacred quest]

I suppose so.

I'm polishing this off before class at 6:40 (PST) so I'll ask my teacher what her thoughts are on the problems when i submit it at the beginning of class

[obsidian jackal]

Can anyone help me how to answer questions like this in webwork?

[bleak thistle]

what did u try

[obsidian jackal]

I tried solving it into a reduced row echelon form

And from there i have x1, x2, x3, and x4 @bleak thistle

I have a unique solution here. I dont understand why is there an s there like i have a parametric solution.

[bleak thistle]

i did not get a unique solution

when i tried it

@obsidian jackal

[obsidian jackal]

Oh

My mistake @bleak thistle

Im still curious what im going to put as an answer in x4

Is it going to be x4=0+1s?

[bleak thistle]

x4 = s

so yeah

[fickle garden]

what is the rank of a matrix?

is that just the dimension of the range of a linear operator?

[dusky epoch]

of its corresponding linear operator, yes